Q- Is there any Hadith that says don’t teach women ‘kitabat’ (to write)?
In the light of what the experts of Hadith have ruled, this Hadith, ‘Do not teach your women kitabat (writing)’ is classified as very weak. Such a Hadith cannot be used to prove anything.(See Al-la’ali al-Masnu’ah: vol. 2 pg. 168; Al-Kamil of Ibn Adi: vol. 2 pg. 395; Ithaful Maharah of Ibn Hajar: vol. 17 pg. 344 and Imdadul Ahkam: vol. 1 pg. 214-215)
Hence, it’s contradiction of other Hadiths will not be taken into consideration.
On the contrary, Imams Abu Dawud and Hakim (rahimahumallah) have recorded the following narration on the authority of Sayyidatuna Al-Shifa bint ‘Abdillah (radiyallahu ‘anha) that she said, ‘Rasulullah (sallallahu ‘alayhi wasallam) entered upon Sayyidatuna Hafsah (radiyallahu ‘anha) and I was with her. So Rasulullah (sallallahu ‘alayhi wasallam)
asked me, ‘Why don’t you teach her the treatment for al-Namlah (an ulcer on the forehead) just as you have taught her to write.’ (Sunan Abi Dawud, Hadith: 3883; Mustadrak Hakim: vol. 4 pg. 56-57)
Imams Hakim and al-Dhahabi (rahimahumallah) have classified this narration as sahih (authentic) according to the standard of Imam Bukhari and Imam Muslim (rahimahumallah). (Ibid)
Allamah Khattabi (rahimahullah) and Hafiz Ibn al-Qayyim (rahimahullah) have stated that this Hadith proves the permissibility of teaching women to write. (Ma’alimus Sunan: vol. 4 pg. 210; Zadul Ma’ad: vol. 4 pg. 170)
Muhaddith Allamah Zafar Ahmad Thanwi (rahimahullah) has stated that the Hadith which prohibits teaching women how to write is very weak and is even classified as a fabrication by some Muhaddithun, hence it is obvious that it cannot be used to challenge this sahih (authentic) Hadith which proves permissibility.(Imdadul Ahkam: vol. 1 pg. 215)
Moulana ‘Abdul-Hayy Lakhnawi (rahimahullah) has offered a similar argument. He has also cited a few examples of some of the pious, knowledgeable women of the past who learnt and knew how to write.(Majmu‘ah Fatawa Lakhnawi: vol. 1 pgs. 131-136)
Caution: It should be noted that Nabi (sallallahu ‘alayhi wasallam) had instructed one female to teach the other. This should not be misinterprated to justify co-education, or any other wrongful method of acquiring knowledge.
And Allah Ta’ala Knows best.